Page: 28
Section: 1.A
Number: 1
Suppose and are real numbers, not both . Find real numbers and such that

Page: 28
Section: 1.A
Number: 2
Show that is a cube root of (meaning that its cube equals ).

Page: 28
Section: 1.A
Number: 3
Find two distinct square roots of .

Page: 28
Section: 1.A
Number: 4
Show that for all .

Page: 28
Section: 1.A
Number: 5
Show that for all .

Page: 28
Section: 1.A
Number: 6
Show that for all .

Page: 28
Section: 1.A
Number: 7
Show that for every , there exists a unique such that .

Page: 28
Section: 1.A
Number: 8
Show that for every with , there exists a unique such that .

Page: 28
Section: 1.A
Number: 9
Show that for all .

Page: 28
Section: 1.A
Number: 10
Find such that

Page: 28
Section: 1.A
Number: 11
Explain why there does not exist such that

Page: 28
Section: 1.A
Number: 12
Show that for all .

Page: 28
Section: 1.A
Number: 13
Show that for all and all .

Page: 28
Section: 1.A
Number: 14
Show that for all .

Page: 28
Section: 1.A
Number: 15
Show that for all and all .

Page: 28
Section: 1.A
Number: 16
Show that for all and all .

Page: 34
Section: 1.B
Number: 1
Prove that for every .

Page: 34
Section: 1.B
Number: 2
Suppose , and . Prove that or .

Page: 34
Section: 1.B
Number: 3
Suppose . Explain why there exists a unique such that .

Page: 34
Section: 1.B
Number: 4
The empty set is not a vector space. The empty set fails to satisfy only one of the requirements listed in 1.19. Which one?

Page: 34
Section: 1.B
Number: 5
Show that in the definition of a vector space (1.19), the additive inverse condition can be replaced with the condition that Here the on the left side is the number , and the on the right side is the additive identity of . (The phrase “a condition can be replaced” in a definition means that the collection of objects satisfying the definition is unchanged if the original condition is replaced with the new condition.)

Page: 34
Section: 1.B
Number: 6
Let and denote two distinct objects, neither of which is in . Define an addition and scalar multiplication on as you could guess from the notation. Specifically, the sum and product of two real numbers is as usual, and for define Is a vector space over ? Explain.

Page: 41
Section: 1.C
Number: 1
For each of the following subsets of , determine whether it is a subspace of : (a) ; (b) ; © ; (d) .

Page: 41
Section: 1.C
Number: 2
Verify all the assertions in Example 1.35.

Page: 41
Section: 1.C
Number: 3
Show that the set of differentiable real-valued functions on the interval such that is a subspace of .

Page: 41
Section: 1.C
Number: 4
Suppose . Show that the set of continuous real-valued functions on the interval such that is a subspace of if and only if .

Page: 41
Section: 1.C
Number: 5
Is a subspace of the complex vector space ?

Page: 41
Section: 1.C
Number: 6
(a) Is a subspace of ? (b) Is a subspace of ?

Page: 41
Section: 1.C
Number: 7
Give an example of a nonempty subset of such that is closed under addition and under taking additive inverses (meaning whenever ), but is not a subspace of .

Page: 41
Section: 1.C
Number: 8
Give an example of a nonempty subset of such that is closed under scalar multiplication, but is not a subspace of . SECTION 1.C Subspaces

Page: 41
Section: 1.C
Number: 9
A function is called periodic if there exists a positive number such that for all . Is the set of periodic functions from to a subspace of ? Explain.

Page: 41
Section: 1.C
Number: 10
Suppose and are subspaces of . Prove that the intersection is a subspace of .

Page: 41
Section: 1.C
Number: 11
Prove that the intersection of every collection of subspaces of is a subspace of .

Page: 41
Section: 1.C
Number: 12
Prove that the union of two subspaces of is a subspace of if and only if one of the subspaces is contained in the other.

Page: 41
Section: 1.C
Number: 13
Prove that the union of three subspaces of is a subspace of if and only if one of the subspaces contains the other two. [This exercise is surprisingly harder than the previous exercise, possibly because this exercise is not true if we replace with a field containing only two elements.]

Page: 41
Section: 1.C
Number: 14
Verify the assertion in Example 1.38.

Page: 41
Section: 1.C
Number: 15
Suppose is a subspace of . What is ?

Page: 41
Section: 1.C
Number: 16
Is the operation of addition on the subspaces of commutative? In other words, if and are subspaces of , is ?

Page: 41
Section: 1.C
Number: 17
Is the operation of addition on the subspaces of associative? In other words, if are subspaces of , is

Page: 41
Section: 1.C
Number: 18
Does the operation of addition on the subspaces of have an additive identity? Which subspaces have additive inverses?

Page: 41
Section: 1.C
Number: 19
Prove or give a counterexample: if are subspaces of such that then .

Page: 41
Section: 1.C
Number: 20
Suppose Find a subspace of such that .

Page: 41
Section: 1.C
Number: 26
CHAPTER Vector Spaces

Page: 41
Section: 1.C
Number: 21
Suppose Find a subspace of such that .

Page: 41
Section: 1.C
Number: 22
Suppose Find three subspaces of , none of which equals , such that .

Page: 41
Section: 1.C
Number: 23
Prove or give a counterexample: if are subspaces of such that then .

Page: 41
Section: 1.C
Number: 24
A function is called even if for all . A function is called if for all . Let denote the set of real-valued even functions on and let denote the set of real-valued odd functions on . Show that .

Page: 54
Section: 2.A
Number: 1
Suppose spans . Prove that the list also spans .

Page: 54
Section: 2.A
Number: 2
Verify the assertions in Example 2.18.

Page: 54
Section: 2.A
Number: 3
Find a number such that is not linearly independent in .

Page: 54
Section: 2.A
Number: 4
Verify the assertion in the second bullet point in Example 2.20.

Page: 54
Section: 2.A
Number: 5
(a) Show that if we think of as a vector space over , then the list is linearly independent. (b) Show that if we think of as a vector space over , then the list is linearly dependent.

Page: 54
Section: 2.A
Number: 6
Suppose is linearly independent in . Prove that the list is also linearly independent.

Page: 54
Section: 2.A
Number: 7
Prove or give a counterexample: If is a linearly independent list of vectors in , then is linearly independent.

Page: 54
Section: 2.A
Number: 8
Prove or give a counterexample: If is a linearly independent list of vectors in and with , then is linearly independent.

Page: 54
Section: 2.A
Number: 9
Prove or give a counterexample: If and are linearly independent lists of vectors in , then is linearly independent.

Page: 54
Section: 2.A
Number: 10
Suppose is linearly independent in and . Prove that if is linearly dependent, then .

Page: 54
Section: 2.A
Number: 38
CHAPTER Finite-Dimensional Vector Spaces

Page: 54
Section: 2.A
Number: 11
Suppose is linearly independent in and . Show that is linearly independent if and only if

Page: 54
Section: 2.A
Number: 12
Explain why there does not exist a list of six polynomials that is linearly independent in .

Page: 54
Section: 2.A
Number: 13
Explain why no list of four polynomials spans .

Page: 54
Section: 2.A
Number: 14
Prove that is infinite-dimensional if and only if there is a sequence of vectors in such that is linearly independent for every positive integer .

Page: 54
Section: 2.A
Number: 15
Prove that is infinite-dimensional.

Page: 54
Section: 2.A
Number: 16
Prove that the real vector space of all continuous real-valued functions on the interval is infinite-dimensional.

Page: 54
Section: 2.A
Number: 17
Suppose are polynomials in such that for each . Prove that is not linearly independent in .

Page: 60
Section: 2.B
Number: 1
Find all vector spaces that have exactly one basis.

Page: 60
Section: 2.B
Number: 2
Verify all the assertions in Example 2.28.

Page: 60
Section: 2.B
Number: 3
(a) Let be the subspace of defined by Find a basis of . (b) Extend the basis in part (a) to a basis of . © Find a subspace of such that .

Page: 60
Section: 2.B
Number: 4
(a) Let be the subspace of defined by Find a basis of . (b) Extend the basis in part (a) to a basis of . © Find a subspace of such that .

Page: 60
Section: 2.B
Number: 5
Prove or disprove: there exists a basis of such that none of the polynomials has degree .

Page: 60
Section: 2.B
Number: 6
Suppose is a basis of . Prove that is also a basis of .

Page: 60
Section: 2.B
Number: 7
Prove or give a counterexample: If is a basis of and is a subspace of such that and and , then is a basis of .

Page: 60
Section: 2.B
Number: 8
Suppose and are subspaces of such that . Suppose also that is a basis of and is a basis of . Prove that is a basis of .

Page: 65
Section: 2.C
Number: 1
Suppose is finite-dimensional and is a subspace of such that . Prove that .

Page: 65
Section: 2.C
Number: 2
Show that the subspaces of are precisely , and all lines in through the origin.

Page: 65
Section: 2.C
Number: 3
Show that the subspaces of are precisely , all lines in through the origin, and all planes in through the origin.

Page: 65
Section: 2.C
Number: 4
(a) Let . Find a basis of . (b) Extend the basis in part (a) to a basis of . © Find a subspace of such that .

Page: 65
Section: 2.C
Number: 5
(a) Let . Find a basis of . (b) Extend the basis in part (a) to a basis of . © Find a subspace of such that .

Page: 65
Section: 2.C
Number: 6
(a) Let . Find a basis of . (b) Extend the basis in part (a) to a basis of . © Find a subspace of such that .

Page: 65
Section: 2.C
Number: 7
(a) Let . Find a basis of . (b) Extend the basis in part (a) to a basis of . © Find a subspace of such that .

Page: 65
Section: 2.C
Number: 8
(a) Let . Find a basis of . (b) Extend the basis in part (a) to a basis of . © Find a subspace of such that .

Page: 65
Section: 2.C
Number: 9
Suppose is linearly independent in and . Prove that .

Page: 65
Section: 2.C
Number: 10
Suppose are such that each has degree . Prove that is a basis of .

Page: 65
Section: 2.C
Number: 11
Suppose that and are subspaces of such that , , and . Prove that . SECTION 2.C Dimension

Page: 65
Section: 2.C
Number: 12
Suppose and are both five-dimensional subspaces of . Prove that .

Page: 65
Section: 2.C
Number: 13
Suppose and are both 4-dimensional subspaces of . Prove that there exist two vectors in such that neither of these vectors is a scalar multiple of the other.

Page: 65
Section: 2.C
Number: 14
Suppose are finite-dimensional subspaces of . Prove that is finite-dimensional and

Page: 65
Section: 2.C
Number: 15
Suppose is finite-dimensional, with . Prove that there exist 1-dimensional subspaces of such that

Page: 65
Section: 2.C
Number: 16
Suppose are finite-dimensional subspaces of such that is a direct sum. Prove that is finitedimensional and [The exercise above deepens the analogy between direct sums of subspaces and disjoint unions of subsets. Specifically, compare this exercise to the following obvious statement: if a set is written as a disjoint union of finite subsets, then the number of elements in the set equals the sum of the numbers of elements in the disjoint subsets.]

Page: 65
Section: 2.C
Number: 17
You might guess, by analogy with the formula for the number of elements in the union of three subsets of a finite set, that if are subspaces of a finite-dimensional vector space, then Prove this or give a counterexample.

Page: 73
Section: 3.A
Number: 1
Suppose . Define by Show that is linear if and only if .

Page: 73
Section: 3.A
Number: 2
Suppose . Define by Show that is linear if and only if .

Page: 73
Section: 3.A
Number: 3
Suppose . Show that there exist scalars for and such that for every . [The exercise above shows that has the form promised in the last item of Example 3.4.]

Page: 73
Section: 3.A
Number: 4
Suppose and is a list of vectors in such that is a linearly independent list in . Prove that is linearly independent.

Page: 73
Section: 3.A
Number: 5
Prove the assertion in 3.7.

Page: 73
Section: 3.A
Number: 6
Prove the assertions in 3.9.

Page: 73
Section: 3.A
Number: 58
CHAPTER Linear Maps

Page: 73
Section: 3.A
Number: 7
Show that every linear map from a 1-dimensional vector space to itself is multiplication by some scalar. More precisely, prove that if and , then there exists such that for all .

Page: 73
Section: 3.A
Number: 8
Give an example of a function such that for all and all but is not linear. [The exercise above and the next exercise show that neither homogeneity nor additivity alone is enough to imply that a function is a linear map.]

Page: 73
Section: 3.A
Number: 9
Give an example of a function such that for all but is not linear. (Here is thought of as a complex vector space.) [There also exists a function such that satisfies the additivity condition above but is not linear. However, showing the existence of such a function involves considerably more advanced tools.]

Page: 73
Section: 3.A
Number: 10
Suppose is a subspace of with . Suppose and (which means that for some ). Define by Prove that is not a linear map on .

Page: 73
Section: 3.A
Number: 11
Suppose is finite-dimensional. Prove that every linear map on a subspace of can be extended to a linear map on . In other words, show that if is a subspace of and , then there exists such that for all .

Page: 73
Section: 3.A
Number: 12
Suppose is finite-dimensional with , and suppose is infinite-dimensional. Prove that is infinite-dimensional.

Page: 73
Section: 3.A
Number: 13
Suppose is a linearly dependent list of vectors in . Suppose also that . Prove that there exist such that no satisfies for each .

Page: 73
Section: 3.A
Number: 14
Suppose is finite-dimensional with . Prove that there exist such that .

Page: 83
Section: 3.B
Number: 1
Give an example of a linear map such that null and dim range .

Page: 83
Section: 3.B
Number: 2
Suppose is a vector space and are such that Prove that .

Page: 83
Section: 3.B
Number: 3
Suppose is a list of vectors in . Define by (a) What property of corresponds to spanning ? (b) What property of corresponds to being linearly independent?

Page: 83
Section: 3.B
Number: 4
Show that is not a subspace of .

Page: 83
Section: 3.B
Number: 5
Give an example of a linear map such that

Page: 83
Section: 3.B
Number: 6
Prove that there does not exist a linear map such that

Page: 83
Section: 3.B
Number: 7
Suppose and are finite-dimensional with . Show that is not injective is not a subspace of .

Page: 83
Section: 3.B
Number: 8
Suppose and are finite-dimensional with . Show that is not surjective is not a subspace of .

Page: 83
Section: 3.B
Number: 9
Suppose is injective and is linearly independent in . Prove that is linearly independent in .

Page: 83
Section: 3.B
Number: 68
CHAPTER Linear Maps

Page: 83
Section: 3.B
Number: 10
Suppose spans and . Prove that the list spans range .

Page: 83
Section: 3.B
Number: 11
Suppose are injective linear maps such that makes sense. Prove that is injective.

Page: 83
Section: 3.B
Number: 12
Suppose that is finite-dimensional and that . Prove that there exists a subspace of such that null and range .

Page: 83
Section: 3.B
Number: 13
Suppose is a linear map from to such that Prove that is surjective.

Page: 83
Section: 3.B
Number: 14
Suppose is a 3-dimensional subspace of and that is a linear map from to such that null . Prove that is surjective.

Page: 83
Section: 3.B
Number: 15
Prove that there does not exist a linear map from to whose null space equals

Page: 83
Section: 3.B
Number: 16
Suppose there exists a linear map on whose null space and range are both finite-dimensional. Prove that is finite-dimensional.

Page: 83
Section: 3.B
Number: 17
Suppose and are both finite-dimensional. Prove that there exists an injective linear map from to if and only if .

Page: 83
Section: 3.B
Number: 18
Suppose and are both finite-dimensional. Prove that there exists a surjective linear map from onto if and only if .

Page: 83
Section: 3.B
Number: 19
Suppose and are finite-dimensional and that is a subspace of . Prove that there exists such that null if and only if .

Page: 83
Section: 3.B
Number: 20
Suppose is finite-dimensional and . Prove that is injective if and only if there exists such that is the identity map on .

Page: 83
Section: 3.B
Number: 21
Suppose is finite-dimensional and . Prove that is surjective if and only if there exists such that is the identity map on . SECTION 3.B Null Spaces and Ranges

Page: 83
Section: 3.B
Number: 22
Suppose and are finite-dimensional vector spaces and and . Prove that

Page: 83
Section: 3.B
Number: 23
Suppose and are finite-dimensional vector spaces and and . Prove that

Page: 83
Section: 3.B
Number: 24
Suppose is finite-dimensional and . Prove that null null if and only if there exists such that .

Page: 83
Section: 3.B
Number: 25
Suppose is finite-dimensional and . Prove that range range if and only if there exists such that .

Page: 83
Section: 3.B
Number: 26
Suppose is such that for every nonconstant polynomial . Prove that is surjective. [The notation is used above to remind you of the differentiation map that sends a polynomial to . Without knowing the formula for the derivative of a polynomial (except that it reduces the degree by 1), you can use the exercise above to show that for every polynomial , there exists a polynomial such that .]

Page: 83
Section: 3.B
Number: 27
Suppose . Prove that there exists a polynomial such that . [This exercise can be done without linear algebra, but it’s more fun to do it using linear algebra.]

Page: 83
Section: 3.B
Number: 28
Suppose , and is a basis of range . Prove that there exist such that for every .

Page: 83
Section: 3.B
Number: 29
Suppose . Suppose is not in null . Prove that

Page: 83
Section: 3.B
Number: 30
Suppose and are linear maps from to that have the same null space. Show that there exists a constant such that .

Page: 83
Section: 3.B
Number: 31
Give an example of two linear maps and from to that have the same null space but are such that is not a scalar multiple of .

Page: 94
Section: 3.C
Number: 1
Suppose and are finite-dimensional and . Show that with respect to each choice of bases of and , the matrix of has at least dim range nonzero entries.

Page: 94
Section: 3.C
Number: 2
Suppose is the differentiation map defined by . Find a basis of and a basis of such that the matrix of with respect to these bases is [Compare the exercise above to Example 3.34. The next exercise generalizes the exercise above.]

Page: 94
Section: 3.C
Number: 3
Suppose and are finite-dimensional and . Prove that there exist a basis of and a basis of such that with respect to these bases, all entries of are except that the entries in row , column , equal for .

Page: 94
Section: 3.C
Number: 4
Suppose is a basis of and is finite-dimensional. Suppose . Prove that there exists a basis of such that all the entries in the first column of (with respect to the bases and are except for possibly a in the first row, first column. [In this exercise, unlike Exercise 3, you are given the basis of instead of being able to choose a basis of .]

Page: 94
Section: 3.C
Number: 5
Suppose is a basis of and is finite-dimensional. Suppose . Prove that there exists a basis of such that all the entries in the first row of (with respect to the bases and are except for possibly a in the first row, first column. [In this exercise, unlike Exercise 3, you are given the basis of instead of being able to choose a basis of .] SECTION 3.C Matrices

Page: 94
Section: 3.C
Number: 6
Suppose and are finite-dimensional and . Prove that dim range if and only if there exist a basis of and a basis of such that with respect to these bases, all entries of equal .

Page: 94
Section: 3.C
Number: 7
Verify .

Page: 94
Section: 3.C
Number: 8
Verify .

Page: 94
Section: 3.C
Number: 9
Prove 3.52.

Page: 94
Section: 3.C
Number: 10
Suppose is an -by- matrix and is an -by- matrix. Prove that for . In other words, show that row of equals (row of ) times .

Page: 94
Section: 3.C
Number: 11
Suppose is a 1-by- matrix and is an -by- matrix. Prove that In other words, show that is a linear combination of the rows of , with the scalars that multiply the rows coming from .

Page: 94
Section: 3.C
Number: 12
Give an example with 2-by-2 matrices to show that matrix multiplication is not commutative. In other words, find 2-by-2 matrices and such that .

Page: 94
Section: 3.C
Number: 13
Prove that the distributive property holds for matrix addition and matrix multiplication. In other words, suppose , and are matrices whose sizes are such that and make sense. Prove that and both make sense and that and .

Page: 94
Section: 3.C
Number: 14
Prove that matrix multiplication is associative. In other words, suppose , and are matrices whose sizes are such that makes sense. Prove that makes sense and that .

Page: 94
Section: 3.C
Number: 15
Suppose is an -by- matrix and . Show that the entry in row , column , of (which is defined to mean ) is

Page: 104
Section: 3.D
Number: 1
Suppose and are both invertible linear maps. Prove that is invertible and that .

Page: 104
Section: 3.D
Number: 2
Suppose is finite-dimensional and . Prove that the set of noninvertible operators on is not a subspace of .

Page: 104
Section: 3.D
Number: 3
Suppose is finite-dimensional, is a subspace of , and . Prove there exists an invertible operator such that for every if and only if is injective.

Page: 104
Section: 3.D
Number: 4
Suppose is finite-dimensional and . Prove that null null if and only if there exists an invertible operator such that .

Page: 104
Section: 3.D
Number: 5
Suppose is finite-dimensional and . Prove that range range if and only if there exists an invertible operator such that .

Page: 104
Section: 3.D
Number: 6
Suppose and are finite-dimensional and . Prove that there exist invertible operators and such that if and only if null null .

Page: 104
Section: 3.D
Number: 7
Suppose and are finite-dimensional. Let . Let (a) Show that is a subspace of . (b) Suppose . What is ? SECTION 3.D Invertibility and Isomorphic Vector Spaces

Page: 104
Section: 3.D
Number: 8
Suppose is finite-dimensional and is a surjective linear map of onto . Prove that there is a subspace of such that is an isomorphism of onto . (Here means the function restricted to . In other words, is the function whose domain is , with defined by for every .)

Page: 104
Section: 3.D
Number: 9
Suppose is finite-dimensional and . Prove that is invertible if and only if both and are invertible.

Page: 104
Section: 3.D
Number: 10
Suppose is finite-dimensional and . Prove that if and only if .

Page: 104
Section: 3.D
Number: 11
Suppose is finite-dimensional and and . Show that is invertible and that .

Page: 104
Section: 3.D
Number: 12
Show that the result in the previous exercise can fail without the hypothesis that is finite-dimensional.

Page: 104
Section: 3.D
Number: 13
Suppose is a finite-dimensional vector space and are such that is surjective. Prove that is injective.

Page: 104
Section: 3.D
Number: 14
Suppose is a basis of . Prove that the map defined by is an isomorphism of onto ; here is the matrix of with respect to the basis .

Page: 104
Section: 3.D
Number: 15
Prove that every linear map from to is given by a matrix multiplication. In other words, prove that if , then there exists an -by- matrix such that for every .

Page: 104
Section: 3.D
Number: 16
Suppose is finite-dimensional and . Prove that is a scalar multiple of the identity if and only if for every .

Page: 104
Section: 3.D
Number: 17
Suppose is finite-dimensional and is a subspace of such that and for all and all . Prove that or .

Page: 104
Section: 3.D
Number: 18
Show that and are isomorphic vector spaces.

Page: 104
Section: 3.D
Number: 19
Suppose is such that is injective and for every nonzero polynomial . (a) Prove that is surjective. (b) Prove that for every nonzero .

Page: 104
Section: 3.D
Number: 90
CHAPTER Linear Maps

Page: 104
Section: 3.D
Number: 20
Suppose is a positive integer and for . Prove that the following are equivalent (note that in both parts below, the number of equations equals the number of variables): (a) The trivial solution is the only solution to the homogeneous system of equations (b) For every , there exists a solution to the system of equations

Page: 114
Section: 3.E
Number: 1
Suppose is a function from to . The graph of is the subset of defined by Prove that is a linear map if and only if the graph of is a subspace of . [Formally, a function from to is a subset of such that for each , there exists exactly one element . In other words, formally a function is what is called above its graph. We do not usually think of functions in this formal manner. However, if we do become formal, then the exercise above could be rephrased as follows: Prove that a function from to is a linear map if and only if is a subspace of .] SECTION 3.E Products and Quotients of Vector Spaces

Page: 114
Section: 3.E
Number: 2
Suppose are vector spaces such that is finitedimensional. Prove that is finite-dimensional for each .

Page: 114
Section: 3.E
Number: 3
Give an example of a vector space and subspaces of such that is isomorphic to but is not a direct sum. Hint: The vector space must be infinite-dimensional.

Page: 114
Section: 3.E
Number: 4
Suppose are vector spaces. Prove that and are isomorphic vector spaces.

Page: 114
Section: 3.E
Number: 5
Suppose are vector spaces. Prove that and are isomorphic vector spaces.

Page: 114
Section: 3.E
Number: 6
For a positive integer, define by Prove that and are isomorphic vector spaces.

Page: 114
Section: 3.E
Number: 7
Suppose are vectors in and are subspaces of such that . Prove that .

Page: 114
Section: 3.E
Number: 8
Prove that a nonempty subset of is an affine subset of if and only if for all and all .

Page: 114
Section: 3.E
Number: 9
Suppose and are affine subsets of . Prove that the intersection is either an affine subset of or the empty set.

Page: 114
Section: 3.E
Number: 10
Prove that the intersection of every collection of affine subsets of is either an affine subset of or the empty set.

Page: 114
Section: 3.E
Number: 11
Suppose . Let (a) Prove that is an affine subset of . (b) Prove that every affine subset of that contains also contains . © Prove that for some and some subspace of with .

Page: 114
Section: 3.E
Number: 12
Suppose is a subspace of such that is finite-dimensional. Prove that is isomorphic to .

Page: 114
Section: 3.E
Number: 100
CHAPTER Linear Maps

Page: 114
Section: 3.E
Number: 13
Suppose is a subspace of and is a basis of and is a basis of . Prove that is a basis of .

Page: 114
Section: 3.E
Number: 14
Suppose for only finitely many . (a) Show that is a subspace of . (b) Prove that is infinite-dimensional.

Page: 114
Section: 3.E
Number: 15
Suppose and . Prove that .

Page: 114
Section: 3.E
Number: 16
Suppose is a subspace of such that . Prove that there exists such that null .

Page: 114
Section: 3.E
Number: 17
Suppose is a subspace of such that is finite-dimensional. Prove that there exists a subspace of such that and .

Page: 114
Section: 3.E
Number: 18
Suppose and is a subspace of . Let denote the quotient map from onto . Prove that there exists such that if and only if null .

Page: 114
Section: 3.E
Number: 19
Find a correct statement analogous to that is applicable to finite sets, with unions analogous to sums of subspaces and disjoint unions analogous to direct sums.

Page: 114
Section: 3.E
Number: 20
Suppose is a subspace of . Define by (a) Show that is a linear map. (b) Show that is injective. © Show that range for every .

Page: 129
Section: 3.F
Number: 1
Explain why every linear functional is either surjective or the zero map.

Page: 129
Section: 3.F
Number: 2
Give three distinct examples of linear functionals on .

Page: 129
Section: 3.F
Number: 3
Suppose is finite-dimensional and with . Prove that there exists such that .

Page: 129
Section: 3.F
Number: 4
Suppose is finite-dimensional and is a subspace of such that . Prove that there exists such that for every but .

Page: 129
Section: 3.F
Number: 5
Suppose are vector spaces. Prove that and are isomorphic vector spaces.

Page: 129
Section: 3.F
Number: 6
Suppose is finite-dimensional and . Define a linear by (a) Prove that spans if and only if is injective. (b) Prove that is linearly independent if and only if is surjective. Suppose is a positive integer. Show that the dual basis of the basis of is , where . Here denotes the derivative of , with the understanding that the derivative of is .

Page: 129
Section: 3.F
Number: 8
Suppose is a positive integer. (a) Show that is a basis of . (b) What is the dual basis of the basis in part (a)?

Page: 129
Section: 3.F
Number: 9
Suppose is a basis of and is the corresponding dual basis of . Suppose . Prove that

Page: 129
Section: 3.F
Number: 10
Prove the first two bullet points in 3.101.

Page: 129
Section: 3.F
Number: 114
CHAPTER Linear Maps

Page: 129
Section: 3.F
Number: 11
Suppose is an -by- matrix with . Prove that the rank of is if and only if there exist and such that for every and every .

Page: 129
Section: 3.F
Number: 12
Show that the dual map of the identity map on is the identity map on .

Page: 129
Section: 3.F
Number: 13
Define by . Suppose denotes the dual basis of the standard basis of and denotes the dual basis of the standard basis of . (a) Describe the linear functionals and . (b) Write and as linear combinations of .

Page: 129
Section: 3.F
Number: 14
Define by for . (a) Suppose is defined by . Describe the linear functional on . (b) Suppose is defined by . Evaluate .

Page: 129
Section: 3.F
Number: 15
Suppose is finite-dimensional and . Prove that if and only if .

Page: 129
Section: 3.F
Number: 16
Suppose and are finite-dimensional. Prove that the map that takes to is an isomorphism of onto .

Page: 129
Section: 3.F
Number: 17
Suppose . Explain why null .

Page: 129
Section: 3.F
Number: 18
Suppose is finite-dimensional and . Show that if and only if .

Page: 129
Section: 3.F
Number: 19
Suppose is finite-dimensional and is a subspace of . Show that if and only if .

Page: 129
Section: 3.F
Number: 20
Suppose and are subsets of with . Prove that .

Page: 129
Section: 3.F
Number: 21
Suppose is finite-dimensional and and are subspaces of with . Prove that .

Page: 129
Section: 3.F
Number: 22
Suppose are subspaces of . Show that . SECTION 3.F Duality

Page: 129
Section: 3.F
Number: 23
Suppose is finite-dimensional and and are subspaces of . Prove that .

Page: 129
Section: 3.F
Number: 24
Prove using the ideas sketched in the discussion before the statement of .

Page: 129
Section: 3.F
Number: 25
Suppose is finite-dimensional and is a subspace of . Show that

Page: 129
Section: 3.F
Number: 26
Suppose is finite-dimensional and is a subspace of . Show that

Page: 129
Section: 3.F
Number: 27
Suppose and null , where is the linear functional on defined by . Prove that range .

Page: 129
Section: 3.F
Number: 28
Suppose and are finite-dimensional, , and there exists such that null . Prove that range .

Page: 129
Section: 3.F
Number: 29
Suppose and are finite-dimensional, , and there exists such that range . Prove that null .

Page: 129
Section: 3.F
Number: 30
Suppose is finite-dimensional and is a linearly independent list in . Prove that

Page: 129
Section: 3.F
Number: 31
Suppose is finite-dimensional and is a basis of . Show that there exists a basis of whose dual basis is .

Page: 129
Section: 3.F
Number: 32
Suppose , and and are bases of . Prove that the following are equivalent: (a) is invertible. (b) The columns of are linearly independent in . © The columns of . (d) The rows of are linearly independent in . (e) The rows of . Here means .

Page: 129
Section: 3.F
Number: 116
CHAPTER Linear Maps

Page: 129
Section: 3.F
Number: 33
Suppose and are positive integers. Prove that the function that takes to is a linear map from to . Furthermore, prove that this linear map is invertible.

Page: 129
Section: 3.F
Number: 34
The double dual space of , denoted , is defined to be the dual space of . In other words, . Define by for and . (a) Show that is a linear map from to . (b) Show that if , then , where . © Show that if is finite-dimensional, then is an isomorphism from onto . [Suppose is finite-dimensional. Then and are isomorphic, but finding an isomorphism from onto generally requires choosing a basis of . In contrast, the isomorphism from onto does not require a choice of basis and thus is considered more natural.]

Page: 129
Section: 3.F
Number: 35
Show that and are isomorphic.

Page: 129
Section: 3.F
Number: 36
Suppose is a subspace of . Let be the inclusion map defined by . Thus . (a) Show that null . (b) Prove that if is finite-dimensional, then range . © Prove that if is finite-dimensional, then is an isomorphism from onto . [The isomorphism in part © is natural in that it does not depend on a choice of basis in either vector space.]

Page: 129
Section: 3.F
Number: 37
Suppose is a subspace of . Let be the usual quotient map. Thus . (a) Show that is injective. (b) Show that range . © Conclude that is an isomorphism from onto . [The isomorphism in part © is natural in that it does not depend on a choice of basis in either vector space. In fact, there is no assumption here that any of these vector spaces are finite-dimensional.]

Page: 145
Section: 4
Number: 1
Verify all the assertions in except the last one.

Page: 145
Section: 4
Number: 2
Suppose is a positive integer. Is the set a subspace of ?

Page: 145
Section: 4
Number: 3
Is the set a subspace of ?

Page: 145
Section: 4
Number: 4
Suppose and are positive integers with , and suppose . Prove that there exists a polynomial with such that and such that has no other zeros.

Page: 145
Section: 4
Number: 5
Suppose is a nonnegative integer, are distinct elements of , and . Prove that there exists a unique polynomial such that for . [This result can be proved without using linear algebra. However, try to find the clearer, shorter proof that uses some linear algebra.]

Page: 145
Section: 4
Number: 6
Suppose has degree . Prove that has distinct zeros if and only if and its derivative have no zeros in common.

Page: 145
Section: 4
Number: 7
Prove that every polynomial of odd degree with real coefficients has a real zero.

Page: 145
Section: 4
Number: 8
Define by Show that for every polynomial and that is a linear map.

Page: 145
Section: 4
Number: 130
CHAPTER Polynomials

Page: 145
Section: 4
Number: 9
Suppose . Define by Prove that is a polynomial with real coefficients.

Page: 145
Section: 4
Number: 10
Suppose is a nonnegative integer and is such that there exist distinct real numbers such that for . Prove that all the coefficients of are real.

Page: 145
Section: 4
Number: 11
Suppose with . Let . (a) Show that . (b) Find a basis of .

Page: 154
Section: 5.A
Number: 1
Suppose and is a subspace of . (a) Prove that if null , then is invariant under . (b) Prove that if range , then is invariant under .

Page: 154
Section: 5.A
Number: 2
Suppose are such that . Prove that null is invariant under . SECTION 5.A Invariant Subspaces

Page: 154
Section: 5.A
Number: 3
Suppose are such that . Prove that range is invariant under .

Page: 154
Section: 5.A
Number: 4
Suppose that and are subspaces of invariant under . Prove that is invariant under .

Page: 154
Section: 5.A
Number: 5
Suppose . Prove that the intersection of every collection of subspaces of invariant under is invariant under .

Page: 154
Section: 5.A
Number: 6
Prove or give a counterexample: if is finite-dimensional and is a subspace of that is invariant under every operator on , then or .

Page: 154
Section: 5.A
Number: 7
Suppose is defined by . Find the eigenvalues of .

Page: 154
Section: 5.A
Number: 8
Define by Find all eigenvalues and eigenvectors of .

Page: 154
Section: 5.A
Number: 9
Define by Find all eigenvalues and eigenvectors of .

Page: 154
Section: 5.A
Number: 10
Define by (a) Find all eigenvalues and eigenvectors of . (b) Find all invariant subspaces of .

Page: 154
Section: 5.A
Number: 11
Define by . Find all eigenvalues and eigenvectors of .

Page: 154
Section: 5.A
Number: 12
Define by for all . Find all eigenvalues and eigenvectors of .

Page: 154
Section: 5.A
Number: 13
Suppose is finite-dimensional, , and . Prove that there exists such that and is invertible.

Page: 154
Section: 5.A
Number: 140
CHAPTER Eigenvalues, Eigenvectors, and Invariant Subspaces

Page: 154
Section: 5.A
Number: 14
Suppose , where and are nonzero subspaces of . Define by for and . Find all eigenvalues and eigenvectors of .

Page: 154
Section: 5.A
Number: 15
Suppose . Suppose is invertible. (a) Prove that and have the same eigenvalues. (b) What is the relationship between the eigenvectors of and the eigenvectors of ?

Page: 154
Section: 5.A
Number: 16
Suppose is a complex vector space, , and the matrix of with respect to some basis of contains only real entries. Show that if is an eigenvalue of , then so is .

Page: 154
Section: 5.A
Number: 17
Give an example of an operator such that has no (real) eigenvalues.

Page: 154
Section: 5.A
Number: 18
Show that the operator defined by has no eigenvalues.

Page: 154
Section: 5.A
Number: 19
Suppose is a positive integer and is defined by in other words, is the operator whose matrix (with respect to the standard basis) consists of all 1’s. Find all eigenvalues and eigenvectors of .

Page: 154
Section: 5.A
Number: 20
Find all eigenvalues and eigenvectors of the backward shift operator defined by

Page: 154
Section: 5.A
Number: 21
Suppose is invertible. (a) Suppose with . Prove that is an eigenvalue of if and only if is an eigenvalue of . (b) Prove that and have the same eigenvectors. SECTION 5.A Invariant Subspaces

Page: 154
Section: 5.A
Number: 22
Suppose and there exist nonzero vectors and in such that Prove that or is an eigenvalue of .

Page: 154
Section: 5.A
Number: 23
Suppose is finite-dimensional and . Prove that and have the same eigenvalues.

Page: 154
Section: 5.A
Number: 24
Suppose is an -by- matrix with entries in . Define by , where elements of are thought of as -by- column vectors. (a) Suppose the sum of the entries in each row of equals . Prove that is an eigenvalue of . (b) Suppose the sum of the entries in each column of equals . Prove that is an eigenvalue of .

Page: 154
Section: 5.A
Number: 25
Suppose and are eigenvectors of such that is also an eigenvector of . Prove that and are eigenvectors of corresponding to the same eigenvalue.

Page: 154
Section: 5.A
Number: 26
Suppose is such that every nonzero vector in is an eigenvector of . Prove that is a scalar multiple of the identity operator.

Page: 154
Section: 5.A
Number: 27
Suppose is finite-dimensional and is such that every subspace of with dimension is invariant under . Prove that is a scalar multiple of the identity operator.

Page: 154
Section: 5.A
Number: 28
Suppose is finite-dimensional with and is such that every 2-dimensional subspace of is invariant under . Prove that is a scalar multiple of the identity operator.

Page: 154
Section: 5.A
Number: 29
Suppose and dim range . Prove that has at most distinct eigenvalues.

Page: 154
Section: 5.A
Number: 30
Suppose and , and are eigenvalues of . Prove that there exists such that .

Page: 154
Section: 5.A
Number: 31
Suppose is finite-dimensional and is a list of vectors in . Prove that is linearly independent if and only if there exists such that are eigenvectors of corresponding to distinct eigenvalues.

Page: 154
Section: 5.A
Number: 142
CHAPTER Eigenvalues, Eigenvectors, and Invariant Subspaces

Page: 154
Section: 5.A
Number: 32
Suppose is a list of distinct real numbers. Prove that the list is linearly independent in the vector space of real-valued functions on . Hint: Let , and define an operator by . Find eigenvalues and eigenvectors of .

Page: 154
Section: 5.A
Number: 33
Suppose . Prove that range .

Page: 154
Section: 5.A
Number: 34
Suppose . Prove that null ) is injective if and only if (null range .

Page: 154
Section: 5.A
Number: 35
Suppose is finite-dimensional, , and is invariant under . Prove that each eigenvalue of is an eigenvalue of . [The exercise below asks you to verify that the hypothesis that is finite-dimensional is needed for the exercise above.]

Page: 154
Section: 5.A
Number: 36
Give an example of a vector space , an operator , and a subspace of that is invariant under such that has an eigenvalue that is not an eigenvalue of .

Page: 169
Section: 5.B
Number: 1
Suppose and there exists a positive integer such that . (a) Prove that is invertible and that (b) Explain how you would guess the formula above.

Page: 169
Section: 5.B
Number: 2
Suppose and . Suppose is an eigenvalue of . Prove that or or .

Page: 169
Section: 5.B
Number: 3
Suppose and and is not an eigenvalue of . Prove that .

Page: 169
Section: 5.B
Number: 4
Suppose and . Prove that null range .

Page: 169
Section: 5.B
Number: 5
Suppose and is invertible. Suppose is a polynomial. Prove that

Page: 169
Section: 5.B
Number: 6
Suppose and is a subspace of invariant under . Prove that is invariant under for every polynomial .

Page: 169
Section: 5.B
Number: 7
Suppose . Prove that is an eigenvalue of if and only if or is an eigenvalue of .

Page: 169
Section: 5.B
Number: 8
Give an example of such that .

Page: 169
Section: 5.B
Number: 9
Suppose is finite-dimensional, , and with . Let be a nonzero polynomial of smallest degree such that . Prove that every zero of is an eigenvalue of .

Page: 169
Section: 5.B
Number: 10
Suppose and is an eigenvector of with eigenvalue . Suppose . Prove that .

Page: 169
Section: 5.B
Number: 11
Suppose is a polynomial, and . Prove that is an eigenvalue of if and only if for some eigenvalue of .

Page: 169
Section: 5.B
Number: 12
Show that the result in the previous exercise does not hold if is replaced with .

Page: 169
Section: 5.B
Number: 154
CHAPTER Eigenvalues, Eigenvectors, and Invariant Subspaces

Page: 169
Section: 5.B
Number: 13
Suppose is a complex vector space and has no eigenvalues. Prove that every subspace of invariant under is either or infinitedimensional.

Page: 169
Section: 5.B
Number: 14
Give an example of an operator whose matrix with respect to some basis contains only 0’s on the diagonal, but the operator is invertible. [The exercise above and the exercise below show that fails without the hypothesis that an upper-triangular matrix is under consideration.]

Page: 169
Section: 5.B
Number: 15
Give an example of an operator whose matrix with respect to some basis contains only nonzero numbers on the diagonal, but the operator is not invertible.

Page: 169
Section: 5.B
Number: 16
Rewrite the proof of using the linear map that sends to (and use 3.23).

Page: 169
Section: 5.B
Number: 17
Rewrite the proof of using the linear map that sends to (and use 3.23).

Page: 169
Section: 5.B
Number: 18
Suppose is a finite-dimensional complex vector space and . Define a function by Prove that is not a continuous function.

Page: 169
Section: 5.B
Number: 19
Suppose is finite-dimensional with and . Prove that

Page: 169
Section: 5.B
Number: 20
Suppose is a finite-dimensional complex vector space and . Prove that has an invariant subspace of dimension for each .

Page: 176
Section: 5.C
Number: 1
Suppose is diagonalizable. Prove that null range .

Page: 176
Section: 5.C
Number: 2
Prove the converse of the statement in the exercise above or give a counterexample to the converse.

Page: 176
Section: 5.C
Number: 3
Suppose is finite-dimensional and . Prove that the following are equivalent: (a) . (b) . © null range .

Page: 176
Section: 5.C
Number: 4
Give an example to show that the exercise above is false without the hypothesis that is finite-dimensional.

Page: 176
Section: 5.C
Number: 5
Suppose is a finite-dimensional complex vector space and . Prove that is diagonalizable if and only if for every .

Page: 176
Section: 5.C
Number: 6
Suppose is finite-dimensional, has distinct eigenvalues, and has the same eigenvectors as (not necessarily with the same eigenvalues). Prove that .

Page: 176
Section: 5.C
Number: 7
Suppose has a diagonal matrix with respect to some basis of and that . Prove that appears on the diagonal of precisely times.

Page: 176
Section: 5.C
Number: 8
Suppose and . Prove that or is invertible. SECTION 5.C Eigenspaces and Diagonal Matrices

Page: 176
Section: 5.C
Number: 9
Suppose is invertible. Prove that for every with .

Page: 176
Section: 5.C
Number: 10
Suppose that is finite-dimensional and . Let denote the distinct nonzero eigenvalues of . Prove that

Page: 176
Section: 5.C
Number: 11
Verify the assertion in Example 5.40.

Page: 176
Section: 5.C
Number: 12
Suppose each have as eigenvalues. Prove that there exists an invertible operator such that .

Page: 176
Section: 5.C
Number: 13
Find such that and each have as eigenvalues, and have no other eigenvalues, and there does not exist an invertible operator such that .

Page: 176
Section: 5.C
Number: 14
Find such that and are eigenvalues of and such that does not have a diagonal matrix with respect to any basis of .

Page: 176
Section: 5.C
Number: 15
Suppose is such that and are eigenvalues of . Furthermore, suppose does not have a diagonal matrix with respect to any basis of . Prove that there exists such that .

Page: 176
Section: 5.C
Number: 16
The Fibonacci sequence is defined by Define by . (a) Show that for each positive integer . (b) Find the eigenvalues of . © Find a basis of consisting of eigenvectors of . (d) Use the solution to part © to compute . Conclude that for each positive integer . (e) Use part (d) to conclude that for each positive integer , the Fibonacci number is the integer that is closest to

Page: 190
Section: 6.A
Number: 1
Show that the function that takes to is not an inner product on .

Page: 190
Section: 6.A
Number: 2
Show that the function that takes to is not an inner product on .

Page: 190
Section: 6.A
Number: 3
Suppose and . Replace the positivity condition (which states that for all ) in the definition of an inner product (6.3) with the condition that for some . Show that this change in the definition does not change the set of functions from to that are inner products on .

Page: 190
Section: 6.A
Number: 4
Suppose is a real inner product space. (a) Show that for every . (b) Show that if have the same norm, then is orthogonal to . © Use part (b) to show that the diagonals of a rhombus are perpendicular to each other.

Page: 190
Section: 6.A
Number: 5
Suppose is such that for every . Prove that is invertible.

Page: 190
Section: 6.A
Number: 6
Suppose . Prove that if and only if for all .

Page: 190
Section: 6.A
Number: 7
Suppose . Prove that for all if and only if .

Page: 190
Section: 6.A
Number: 8
Suppose and and . Prove that .

Page: 190
Section: 6.A
Number: 9
Suppose and and . Prove that

Page: 190
Section: 6.A
Number: 10
Find vectors such that is a scalar multiple of is orthogonal to , and .

Page: 190
Section: 6.A
Number: 176
CHAPTER Inner Product Spaces

Page: 190
Section: 6.A
Number: 11
Prove that for all positive numbers .

Page: 190
Section: 6.A
Number: 12
Prove that for all positive integers and all real numbers .

Page: 190
Section: 6.A
Number: 13
Suppose are nonzero vectors in . Prove that where is the angle between and (thinking of and as arrows with initial point at the origin). Hint: Draw the triangle formed by , and ; then use the law of cosines.

Page: 190
Section: 6.A
Number: 14
The angle between two vectors (thought of as arrows with initial point at the origin) in or can be defined geometrically. However, geometry is not as clear in for . Thus the angle between two nonzero vectors is defined to be where the motivation for this definition comes from the previous exercise. Explain why the Cauchy-Schwarz Inequality is needed to show that this definition makes sense.

Page: 190
Section: 6.A
Number: 15
Prove that for all real numbers and .

Page: 190
Section: 6.A
Number: 16
Suppose are such that What number does equal? SECTION 6.A Inner Products and Norms

Page: 190
Section: 6.A
Number: 17
Prove or disprove: there is an inner product on such that the associated norm is given by for all .

Page: 190
Section: 6.A
Number: 18
Suppose . Prove that there is an inner product on such that the associated norm is given by for all if and only if .

Page: 190
Section: 6.A
Number: 19
Suppose is a real inner product space. Prove that for all .

Page: 190
Section: 6.A
Number: 20
Suppose is a complex inner product space. Prove that for all .

Page: 190
Section: 6.A
Number: 21
A norm on a vector space is a function || such that if and only if for all and all , and for all . Prove that a norm satisfying the parallelogram equality comes from an inner product (in other words, show that if || is a norm on satisfying the parallelogram equality, then there is an inner product , such that for all .

Page: 190
Section: 6.A
Number: 22
Show that the square of an average is less than or equal to the average of the squares. More precisely, show that if , then the square of the average of is less than or equal to the average of .

Page: 190
Section: 6.A
Number: 23
Suppose are inner product spaces. Show that the equation defines an inner product on . [In the expression above on the right, denotes the inner product on denotes the inner product on . Each of the spaces may have a different inner product, even though the same notation is used here.]

Page: 190
Section: 6.A
Number: 178
CHAPTER Inner Product Spaces

Page: 190
Section: 6.A
Number: 24
Suppose is an injective operator on . Define by for . Show that is an inner product on .

Page: 190
Section: 6.A
Number: 25
Suppose is not injective. Define as in the exercise above. Explain why is not an inner product on .

Page: 190
Section: 6.A
Number: 26
Suppose are differentiable functions from to . (a) Show that (b) Suppose and for every . Show that for every . © Interpret the result in part (b) geometrically in terms of the tangent vector to a curve lying on a sphere in centered at the origin. [For the exercise above, a function is called differentiable if there exist differentiable functions from to such that for each . Furthermore, for each , the derivative is defined by .]

Page: 190
Section: 6.A
Number: 27
Suppose . Prove that

Page: 190
Section: 6.A
Number: 28
Suppose is a subset of with the property that implies . Let . Show that there is at most one point in that is closest to . In other words, show that there is at most one such that Hint: Use the previous exercise.

Page: 190
Section: 6.A
Number: 29
For , define . (a) Show that is a metric on . (b) Show that if is finite-dimensional, then is a complete metric on (meaning that every Cauchy sequence converges). © Show that every finite-dimensional subspace of is a closed subset of (with respect to the metric ). SECTION 6.A Inner Products and Norms

Page: 190
Section: 6.A
Number: 30
Fix a positive integer . The Laplacian of a twice differentiable function on is the function on defined by The function is called harmonic if . A polynomial on is a linear combination of functions of the form , where are nonnegative integers. Suppose is a polynomial on . Prove that there exists a harmonic polynomial on such that for every with . [The only fact about harmonic functions that you need for this exercise is that if is a harmonic function on and for all with , then .] Hint: A reasonable guess is that the desired harmonic polynomial is of the form for some polynomial . Prove that there is a polynomial on such that is harmonic by defining an operator on a suitable vector space by and then showing that is injective and hence surjective.

Page: 190
Section: 6.A
Number: 31
Use inner products to prove Apollonius’s Identity: In a triangle with sides of length , and , let be the length of the line segment from the midpoint of the side of length to the opposite vertex. Then

Page: 204
Section: 6.B
Number: 1
(a) Suppose . Show that and are orthonormal bases of . (b) Show that each orthonormal basis of is of the form given by one of the two possibilities of part (a).

Page: 204
Section: 6.B
Number: 2
Suppose is an orthonormal list of vectors in . Let . Prove that if and only if .

Page: 204
Section: 6.B
Number: 3
Suppose has an upper-triangular matrix with respect to the basis . Find an orthonormal basis of (use the usual inner product on ) with respect to which has an upper-triangular matrix.

Page: 204
Section: 6.B
Number: 190
CHAPTER Inner Product Spaces

Page: 204
Section: 6.B
Number: 4
Suppose is a positive integer. Prove that is an orthonormal list of vectors in , the vector space of continuous real-valued functions on with inner product [The orthonormal list above is often used for modeling periodic phenomena such as tides.]

Page: 204
Section: 6.B
Number: 5
On , consider the inner product given by Apply the Gram-Schmidt Procedure to the basis to produce an orthonormal basis of .

Page: 204
Section: 6.B
Number: 6
Find an orthonormal basis of (with inner product as in Exercise 5) such that the differentiation operator (the operator that takes to ) on has an upper-triangular matrix with respect to this basis.

Page: 204
Section: 6.B
Number: 7
Find a polynomial such that for every .

Page: 204
Section: 6.B
Number: 8
Find a polynomial such that for every .

Page: 204
Section: 6.B
Number: 9
What happens if the Gram-Schmidt Procedure is applied to a list of vectors that is not linearly independent? SECTION 6.B Orthonormal Bases

Page: 204
Section: 6.B
Number: 10
Suppose is a real inner product space and is a linearly independent list of vectors in . Prove that there exist exactly orthonormal lists of vectors in such that for all .

Page: 204
Section: 6.B
Number: 11
Suppose and are inner products on such that if and only if . Prove that there is a positive number such that for every .

Page: 204
Section: 6.B
Number: 12
Suppose is finite-dimensional and are inner products on with corresponding norms and . Prove that there exists a positive number such that for every .

Page: 204
Section: 6.B
Number: 13
Suppose is a linearly independent list in . Show that there exists such that for all .

Page: 204
Section: 6.B
Number: 14
Suppose is an orthonormal basis of and are vectors in such that for each . Prove that is a basis of .

Page: 204
Section: 6.B
Number: 15
Suppose is the vector space of continuous real-valued functions on the interval with inner product given by for . Let be the linear functional on defined by . Show that there does not exist such that for every . [The exercise above shows that the Riesz Representation Theorem (6.42) does not hold on infinite-dimensional vector spaces without additional hypotheses on and .]

Page: 204
Section: 6.B
Number: 192
CHAPTER Inner Product Spaces

Page: 204
Section: 6.B
Number: 16
Suppose is finite-dimensional, , all the eigenvalues of have absolute value less than , and . Prove that there exists a positive integer such that for every .

Page: 204
Section: 6.B
Number: 17
For , let denote the linear functional on defined by for . (a) Show that if , then is a linear map from to . (Recall from Section 3.F that and that is called the dual space of .) (b) Show that if and , then is not a linear map. © Show that is injective. (d) Suppose and is finite-dimensional. Use parts (a) and © and a dimension-counting argument (but without using 6.42) to show that is an isomorphism from onto . [Part (d) gives an alternative proof of the Riesz Representation Theorem (6.42) when . Part (d) also gives a natural isomorphism (meaning that it does not depend on a choice of basis) from a finite-dimensional real inner product space onto its dual space.]

Page: 216
Section: 6.C
Number: 1
Suppose . Prove that

Page: 216
Section: 6.C
Number: 2
Suppose is a finite-dimensional subspace of . Prove that if and only if . [Exercise 14(a) shows that the result above is not true without the hypothesis that is finite-dimensional.]

Page: 216
Section: 6.C
Number: 3
Suppose is a subspace of with basis and is a basis of . Prove that if the Gram-Schmidt Procedure is applied to the basis of above, producing a list , then is an orthonormal basis of and is an orthonormal basis of .

Page: 216
Section: 6.C
Number: 4
Suppose is the subspace of defined by Find an orthonormal basis of and an orthonormal basis of .

Page: 216
Section: 6.C
Number: 5
Suppose is finite-dimensional and is a subspace of . Show that , where is the identity operator on .

Page: 216
Section: 6.C
Number: 6
Suppose and are finite-dimensional subspaces of . Prove that if and only if for all and all .

Page: 216
Section: 6.C
Number: 7
Suppose is finite-dimensional and is such that and every vector in null is orthogonal to every vector in range . Prove that there exists a subspace of such that .

Page: 216
Section: 6.C
Number: 8
Suppose is finite-dimensional and is such that and for every . Prove that there exists a subspace of such that .

Page: 216
Section: 6.C
Number: 9
Suppose and is a finite-dimensional subspace of . Prove that is invariant under if and only if .

Page: 216
Section: 6.C
Number: 202
CHAPTER Inner Product Spaces

Page: 216
Section: 6.C
Number: 10
Suppose is finite-dimensional, , and is a subspace of . Prove that and are both invariant under if and only if .

Page: 216
Section: 6.C
Number: 11
In , let Find such that is as small as possible.

Page: 216
Section: 6.C
Number: 12
Find such that , and is as small as possible.

Page: 216
Section: 6.C
Number: 13
Find that makes as small as possible. [The polynomial is an excellent approximation to the answer to this exercise, but here you are asked to find the exact solution, which involves powers of . A computer that can perform symbolic integration will be useful.]

Page: 216
Section: 6.C
Number: 14
Suppose is the vector space of continuous real-valued functions on the interval with inner product given by for . Let be the subspace of defined by (a) Show that . (b) Show that and do not hold without the finite-dimensional hypothesis.

Page: 229
Section: 7.A
Number: 1
Suppose is a positive integer. Define by Find a formula for .

Page: 229
Section: 7.A
Number: 2
Suppose and . Prove that is an eigenvalue of if and only if is an eigenvalue of .

Page: 229
Section: 7.A
Number: 3
Suppose and is a subspace of . Prove that is invariant under if and only if is invariant under .

Page: 229
Section: 7.A
Number: 4
Suppose . Prove that (a) is injective if and only if is surjective; (b) is surjective if and only if is injective.

Page: 229
Section: 7.A
Number: 5
Prove that and for every . SECTION 7.A Self-Adjoint and Normal Operators

Page: 229
Section: 7.A
Number: 6
Make into an inner product space by defining Define by . (a) Show that is not self-adjoint. (b) The matrix of with respect to the basis is This matrix equals its conjugate transpose, even though is not self-adjoint. Explain why this is not a contradiction.

Page: 229
Section: 7.A
Number: 7
Suppose are self-adjoint. Prove that is self-adjoint if and only if .

Page: 229
Section: 7.A
Number: 8
Suppose is a real inner product space. Show that the set of self-adjoint operators on is a subspace of .

Page: 229
Section: 7.A
Number: 9
Suppose is a complex inner product space with . Show that the set of self-adjoint operators on is not a subspace of .

Page: 229
Section: 7.A
Number: 10
Suppose . Show that the set of normal operators on is not a subspace of .

Page: 229
Section: 7.A
Number: 11
Suppose is such that . Prove that there is a subspace of such that if and only if is self-adjoint.

Page: 229
Section: 7.A
Number: 12
Suppose that is a normal operator on and that and are eigenvalues of . Prove that there exists a vector such that and .

Page: 229
Section: 7.A
Number: 13
Give an example of an operator such that is normal but not self-adjoint.

Page: 229
Section: 7.A
Number: 14
Suppose is a normal operator on . Suppose also that satisfy the equations Show that .

Page: 229
Section: 7.A
Number: 216
CHAPTER Operators on Inner Product Spaces

Page: 229
Section: 7.A
Number: 15
Fix . Define by for every . (a) Suppose . Prove that is self-adjoint if and only if is linearly dependent. (b) Prove that is normal if and only if is linearly dependent.

Page: 229
Section: 7.A
Number: 16
Suppose is normal. Prove that

Page: 229
Section: 7.A
Number: 17
Suppose is normal. Prove that for every positive integer .

Page: 229
Section: 7.A
Number: 18
Prove or give a counterexample: If and there exists an orthonormal basis of such that for each , then is normal.

Page: 229
Section: 7.A
Number: 19
Suppose is normal and . Suppose . Prove that .

Page: 229
Section: 7.A
Number: 20
Suppose and . Let be the isomorphism from onto the dual space given by Exercise in Section 6.B, and let be the corresponding isomorphism from onto . Show that if and are used to identify and with and , then is identified with the dual map . More precisely, show that .

Page: 229
Section: 7.A
Number: 21
Fix a positive integer . In the inner product space of continuous realvalued functions on with inner product let (a) Define by . Show that . Conclude that is normal but not self-adjoint. (b) Define by . Show that is self-adjoint.

Page: 238
Section: 7.B
Number: 1
True or false (and give a proof of your answer): There exists such that is not self-adjoint (with respect to the usual inner product) and such that there is a basis of consisting of eigenvectors of .

Page: 238
Section: 7.B
Number: 2
Suppose that is a self-adjoint operator on a finite-dimensional inner product space and that and are the only eigenvalues of . Prove that .

Page: 238
Section: 7.B
Number: 3
Give an example of an operator such that and are the only eigenvalues of and .

Page: 238
Section: 7.B
Number: 4
Suppose and . Prove that is normal if and only if all pairs of eigenvectors corresponding to distinct eigenvalues of are orthogonal and where denote the distinct eigenvalues of .

Page: 238
Section: 7.B
Number: 5
Suppose and . Prove that is self-adjoint if and only if all pairs of eigenvectors corresponding to distinct eigenvalues of are orthogonal and where denote the distinct eigenvalues of .

Page: 238
Section: 7.B
Number: 6
Prove that a normal operator on a complex inner product space is selfadjoint if and only if all its eigenvalues are real. [The exercise above strengthens the analogy (for normal operators) between self-adjoint operators and real numbers.]

Page: 238
Section: 7.B
Number: 7
Suppose is a complex inner product space and is a normal operator such that . Prove that is self-adjoint and .

Page: 238
Section: 7.B
Number: 8
Give an example of an operator on a complex vector space such that but .

Page: 238
Section: 7.B
Number: 9
Suppose is a complex inner product space. Prove that every normal operator on has a square root. (An operator is called a square root of if .)

Page: 238
Section: 7.B
Number: 224
CHAPTER Operators on Inner Product Spaces

Page: 238
Section: 7.B
Number: 10
Give an example of a real inner product space and and real numbers with such that is not invertible. [The exercise above shows that the hypothesis that is self-adjoint is needed in 7.26, even for real vector spaces.]

Page: 238
Section: 7.B
Number: 11
Prove or give a counterexample: every self-adjoint operator on has a cube root. (An operator is called a cube root of if .)

Page: 238
Section: 7.B
Number: 12
Suppose is self-adjoint, , and . Suppose there exists such that and Prove that has an eigenvalue such that .

Page: 238
Section: 7.B
Number: 13
Give an alternative proof of the Complex Spectral Theorem that avoids Schur’s Theorem and instead follows the pattern of the proof of the Real Spectral Theorem.

Page: 238
Section: 7.B
Number: 14
Suppose is a finite-dimensional real vector space and . Prove that has a basis consisting of eigenvectors of if and only if there is an inner product on that makes into a self-adjoint operator.

Page: 238
Section: 7.B
Number: 15
Find the matrix entry below that is covered up.

Page: 246
Section: 7.C
Number: 1
Prove or give a counterexample: If is self-adjoint and there exists an orthonormal basis of such that for each , then is a positive operator.

Page: 246
Section: 7.C
Number: 2
Suppose is a positive operator on . Suppose are such that Prove that .

Page: 246
Section: 7.C
Number: 3
Suppose is a positive operator on and is a subspace of invariant under . Prove that is a positive operator on .

Page: 246
Section: 7.C
Number: 4
Suppose . Prove that is a positive operator on and is a positive operator on .

Page: 246
Section: 7.C
Number: 232
CHAPTER Operators on Inner Product Spaces

Page: 246
Section: 7.C
Number: 5
Prove that the sum of two positive operators on is positive.

Page: 246
Section: 7.C
Number: 6
Suppose is positive. Prove that is positive for every positive integer .

Page: 246
Section: 7.C
Number: 7
Suppose is a positive operator on . Prove that is invertible if and only if for every with .

Page: 246
Section: 7.C
Number: 8
Suppose . For , define by Prove that is an inner product on if and only if is an invertible positive operator (with respect to the original inner product ).

Page: 246
Section: 7.C
Number: 9
Prove or disprove: the identity operator on has infinitely many selfadjoint square roots.

Page: 246
Section: 7.C
Number: 10
Suppose . Prove that the following are equivalent: (a) is an isometry; (b) for all ; © is an orthonormal list for every orthonormal list of vectors in ; (d) is an orthonormal basis for some orthonormal basis of .

Page: 246
Section: 7.C
Number: 11
Suppose are normal operators on and both operators have 2, 5, as eigenvalues. Prove that there exists an isometry such that .

Page: 246
Section: 7.C
Number: 12
Give an example of two self-adjoint operators such that the eigenvalues of both operators are 2, 5,7 but there does not exist an isometry such that . Be sure to explain why there is no isometry with the required property.

Page: 246
Section: 7.C
Number: 13
Prove or give a counterexample: if and there exists an orthonormal basis of such that for each , then is an isometry.

Page: 246
Section: 7.C
Number: 14
Let be the second derivative operator in Exercise in Section 7.A. Show that is a positive operator.

Page: 253
Section: 7.D
Number: 1
Fix with . Define by for every . Prove that for every .

Page: 253
Section: 7.D
Number: 2
Give an example of such that is the only eigenvalue of and the singular values of are . SECTION 7.D Polar Decomposition and Singular Value Decomposition

Page: 253
Section: 7.D
Number: 3
Suppose . Prove that there exists an isometry such that

Page: 253
Section: 7.D
Number: 4
Suppose and is a singular value of . Prove that there exists a vector such that and .

Page: 253
Section: 7.D
Number: 5
Suppose is defined by . Find the singular values of .

Page: 253
Section: 7.D
Number: 6
Find the singular values of the differentiation operator defined by , where the inner product on is as in Example . Define by Find (explicitly) an isometry such that .

Page: 253
Section: 7.D
Number: 8
Suppose is an isometry, and is a positive operator such that . Prove that . [The exercise above shows that if we write as the product of an isometry and a positive operator (as in the Polar Decomposition 7.45), then the positive operator equals .]

Page: 253
Section: 7.D
Number: 9
Suppose . Prove that is invertible if and only if there exists a unique isometry such that .

Page: 253
Section: 7.D
Number: 10
Suppose is self-adjoint. Prove that the singular values of equal the absolute values of the eigenvalues of , repeated appropriately.

Page: 253
Section: 7.D
Number: 11
Suppose . Prove that and have the same singular values.

Page: 253
Section: 7.D
Number: 12
Prove or give a counterexample: if , then the singular values of equal the squares of the singular values of .

Page: 253
Section: 7.D
Number: 13
Suppose . Prove that is invertible if and only if is not a singular value of .

Page: 253
Section: 7.D
Number: 14
Suppose . Prove that dim range equals the number of nonzero singular values of .

Page: 253
Section: 7.D
Number: 15
Suppose . Prove that is an isometry if and only if all the singular values of equal .

Page: 253
Section: 7.D
Number: 240
CHAPTER Operators on Inner Product Spaces

Page: 253
Section: 7.D
Number: 16
Suppose . Prove that and have the same singular values if and only if there exist isometries such that .

Page: 253
Section: 7.D
Number: 17
Suppose has singular value decomposition given by for every , where are the singular values of and and are orthonormal bases of . (a) Prove that if , then (b) Prove that if , then © Prove that if , then (d) Suppose is invertible. Prove that if , then for every .

Page: 253
Section: 7.D
Number: 18
Suppose . Let denote the smallest singular value of , and let denote the largest singular value of . (a) Prove that for every . (b) Suppose is an eigenvalue of . Prove that .

Page: 253
Section: 7.D
Number: 19
Suppose . Show that is uniformly continuous with respect to the metric on defined by .

Page: 253
Section: 7.D
Number: 20
Suppose . Let denote the largest singular value of , let denote the largest singular value of , and let denote the largest singular value of . Prove that .

Page: 264
Section: 8.A
Number: 1
Define by Find all generalized eigenvectors of .

Page: 264
Section: 8.A
Number: 2
Define by Find the generalized eigenspaces corresponding to the distinct eigenvalues of .

Page: 264
Section: 8.A
Number: 250
CHAPTER Operators on Complex Vector Spaces

Page: 264
Section: 8.A
Number: 3
Suppose is invertible. Prove that for every with .

Page: 264
Section: 8.A
Number: 4
Suppose and with . Prove that

Page: 264
Section: 8.A
Number: 5
Suppose is a positive integer, and is such that but . Prove that is linearly independent.

Page: 264
Section: 8.A
Number: 6
Suppose is defined by . Prove that has no square root. More precisely, prove that there does not exist such that .

Page: 264
Section: 8.A
Number: 7
Suppose is nilpotent. Prove that is the only eigenvalue of .

Page: 264
Section: 8.A
Number: 8
Prove or give a counterexample: The set of nilpotent operators on is a subspace of .

Page: 264
Section: 8.A
Number: 9
Suppose and is nilpotent. Prove that is nilpotent.

Page: 264
Section: 8.A
Number: 10
Suppose that is not nilpotent. Let . Show that .

Page: 264
Section: 8.A
Number: 11
Prove or give a counterexample: If is a complex vector space and and , then is diagonalizable.

Page: 264
Section: 8.A
Number: 12
Suppose and there exists a basis of with respect to which has an upper-triangular matrix with only 's on the diagonal. Prove that is nilpotent.

Page: 264
Section: 8.A
Number: 13
Suppose is an inner product space and is normal and nilpotent. Prove that .

Page: 264
Section: 8.A
Number: 14
Suppose is an inner product space and is nilpotent. Prove that there exists an orthonormal basis of with respect to which has an upper-triangular matrix. [If , then the result above follows from Schur’s Theorem (6.38) without the hypothesis that is nilpotent. Thus the exercise above needs to be proved only when .] SECTION 8.A Generalized Eigenvectors and Nilpotent Operators

Page: 264
Section: 8.A
Number: 15
Suppose is such that null . Prove that is nilpotent and that for every integer with .

Page: 264
Section: 8.A
Number: 16
Suppose . Show that

Page: 264
Section: 8.A
Number: 17
Suppose and is a nonnegative integer such that Prove that range for all .

Page: 264
Section: 8.A
Number: 18
Suppose . Let . Prove that

Page: 264
Section: 8.A
Number: 19
Suppose and is a nonnegative integer. Prove that null if and only if .

Page: 264
Section: 8.A
Number: 20
Suppose is such that range . Prove that is nilpotent.

Page: 264
Section: 8.A
Number: 21
Find a vector space and such that null and range range for every positive integer .

Page: 274
Section: 8.B
Number: 1
Suppose is a complex vector space, , and is the only eigenvalue of . Prove that is nilpotent.

Page: 274
Section: 8.B
Number: 2
Give an example of an operator on a finite-dimensional real vector space such that is the only eigenvalue of but is not nilpotent.

Page: 274
Section: 8.B
Number: 260
CHAPTER Operators on Complex Vector Spaces

Page: 274
Section: 8.B
Number: 3
Suppose . Suppose is invertible. Prove that and have the same eigenvalues with the same multiplicities.

Page: 274
Section: 8.B
Number: 4
Suppose is an -dimensional complex vector space and is an operator on such that null . Prove that has at most two distinct eigenvalues.

Page: 274
Section: 8.B
Number: 5
Suppose is a complex vector space and . Prove that has a basis consisting of eigenvectors of if and only if every generalized eigenvector of is an eigenvector of . [For , the exercise above adds an equivalence to the list in 5.41.]

Page: 274
Section: 8.B
Number: 6
Define by Find a square root of . Suppose is a complex vector space. Prove that every invertible operator on has a cube root.

Page: 274
Section: 8.B
Number: 8
Suppose and and are eigenvalues of . Let . Prove that null range .

Page: 274
Section: 8.B
Number: 9
Suppose and are block diagonal matrices of the form where has the same size as for . Show that is a block diagonal matrix of the form

Page: 274
Section: 8.B
Number: 10
Suppose and . Prove that there exist such that , the operator is diagonalizable, is nilpotent, and .

Page: 274
Section: 8.B
Number: 11
Suppose and . Prove that for every basis of with respect to which has an upper-triangular matrix, the number of times that appears on the diagonal of the matrix of equals the multiplicity of as an eigenvalue of . SECTION 8.C Characteristic and Minimal Polynomials

Page: 274
Section: 8.B
Number: 261
8.C Characteristic and Minimal Polynomials The Cayley-Hamilton Theorem The next definition associates a polynomial with each operator on if . For , the corresponding definition will be given in the next chapter. 8.34 Definition characteristic polynomial Suppose is a complex vector space and . Let denote the distinct eigenvalues of , with multiplicities . The polynomial is called the characteristic polynomial of . 8.35 Example Suppose is defined as in Example 8.25. Because the eigenvalues of are , with multiplicity , and , with multiplicity , we see that the characteristic polynomial of is . 8.36 Degree and zeros of characteristic polynomial Suppose is a complex vector space and . Then (a) the characteristic polynomial of has degree ; (b) the zeros of the characteristic polynomial of are the eigenvalues of . Proof Clearly part (a) follows from and part (b) follows from the definition of the characteristic polynomial. Most texts define the characteristic polynomial using determinants (the two definitions are equivalent by 10.25). The approach taken here, which is considerably simpler, leads to the following easy proof of the CayleyHamilton Theorem. In the next chapter, we will see that this result also holds on real vector spaces (see 9.24). 8.37 Cayley-Hamilton Theorem Suppose is a complex vector space and . Let denote the characteristic polynomial of . Then .

Page: 282
Section: 8.C
Number: 1
Suppose is such that the eigenvalues of are 3, 5, 8. Prove that .

Page: 282
Section: 8.C
Number: 2
Suppose is a complex vector space. Suppose is such that and are eigenvalues of and that has no other eigenvalues. Prove that , where .

Page: 282
Section: 8.C
Number: 3
Give an example of an operator on whose characteristic polynomial equals .

Page: 282
Section: 8.C
Number: 268
CHAPTER Operators on Complex Vector Spaces

Page: 282
Section: 8.C
Number: 4
Give an example of an operator on whose characteristic polynomial equals and whose minimal polynomial equals .

Page: 282
Section: 8.C
Number: 5
Give an example of an operator on whose characteristic and minimal polynomials both equal .

Page: 282
Section: 8.C
Number: 6
Give an example of an operator on whose characteristic polynomial equals and whose minimal polynomial equals .

Page: 282
Section: 8.C
Number: 7
Suppose is a complex vector space. Suppose is such that . Prove that the characteristic polynomial of is , where null and .

Page: 282
Section: 8.C
Number: 8
Suppose . Prove that is invertible if and only if the constant term in the minimal polynomial of is nonzero.

Page: 282
Section: 8.C
Number: 9
Suppose has minimal polynomial . Find the minimal polynomial of .

Page: 282
Section: 8.C
Number: 10
Suppose is a complex vector space and is invertible. Let denote the characteristic polynomial of and let denote the characteristic polynomial of . Prove that for all nonzero .

Page: 282
Section: 8.C
Number: 11
Suppose is invertible. Prove that there exists a polynomial such that .

Page: 282
Section: 8.C
Number: 12
Suppose is a complex vector space and . Prove that has a basis consisting of eigenvectors of if and only if the minimal polynomial of has no repeated zeros. [For complex vector spaces, the exercise above adds another equivalence to the list given by 5.41.]

Page: 282
Section: 8.C
Number: 13
Suppose is an inner product space and is normal. Prove that the minimal polynomial of has no repeated zeros.

Page: 282
Section: 8.C
Number: 14
Suppose is a complex inner product space and is an isometry. Prove that the constant term in the characteristic polynomial of has absolute value . SECTION 8.C Characteristic and Minimal Polynomials

Page: 282
Section: 8.C
Number: 15
Suppose and . (a) Prove that there exists a unique monic polynomial of smallest degree such that . (b) Prove that divides the minimal polynomial of .

Page: 282
Section: 8.C
Number: 16
Suppose is an inner product space and . Suppose is the minimal polynomial of . Prove that is the minimal polynomial of .

Page: 282
Section: 8.C
Number: 17
Suppose and . Suppose the minimal polynomial of has degree . Prove that the characteristic polynomial of equals the minimal polynomial of .

Page: 282
Section: 8.C
Number: 18
Suppose . Find the minimal and characteristic polynomials of the operator on whose matrix (with respect to the standard basis) is [The exercise above shows that every monic polynomial is the characteristic polynomial of some operator.]

Page: 282
Section: 8.C
Number: 19
Suppose is a complex vector space and . Suppose that with respect to some basis of the matrix of is upper triangular, with on the diagonal of this matrix. Prove that the characteristic polynomial of is .

Page: 282
Section: 8.C
Number: 20
Suppose is a complex vector space and are nonzero subspaces of such that . Suppose and each is invariant under . For each , let denote the characteristic polynomial of . Prove that the characteristic polynomial of equals

Page: 289
Section: 8.D
Number: 1
Find the characteristic polynomial and the minimal polynomial of the operator in Example 8.53.

Page: 289
Section: 8.D
Number: 2
Find the characteristic polynomial and the minimal polynomial of the operator in Example 8.54.

Page: 289
Section: 8.D
Number: 3
Suppose is nilpotent. Prove that the minimal polynomial of is , where is the length of the longest consecutive string of 's that appears on the line directly above the diagonal in the matrix of with respect to any Jordan basis for .

Page: 289
Section: 8.D
Number: 4
Suppose and is a basis of that is a Jordan basis for . Describe the matrix of with respect to the basis obtained by reversing the order of the 's.

Page: 289
Section: 8.D
Number: 5
Suppose and is a basis of that is a Jordan basis for . Describe the matrix of with respect to this basis.

Page: 289
Section: 8.D
Number: 6
Suppose is nilpotent and and are as in 8.55. Prove that is a basis of null . [The exercise above implies that , which equals dim null , depends only on and not on the specific Jordan basis chosen for .] Suppose are monic polynomials with the same zeros and is a polynomial multiple of . Prove that there exists such that the characteristic polynomial of is and the minimal polynomial of is .

Page: 289
Section: 8.D
Number: 8
Suppose is a complex vector space and . Prove that there does not exist a direct sum decomposition of into two proper subspaces invariant under if and only if the minimal polynomial of is of the form for some .

Page: 300
Section: 9.A
Number: 1
Prove 9.3.

Page: 300
Section: 9.A
Number: 2
Verify that if is a real vector space and , then .

Page: 300
Section: 9.A
Number: 3
Suppose is a real vector space and . Prove that is linearly independent in if and only if is linearly independent in .

Page: 300
Section: 9.A
Number: 4
Suppose is a real vector space and . Prove that spans if and only if spans .

Page: 300
Section: 9.A
Number: 5
Suppose that is a real vector space and . Show that and that for every .

Page: 300
Section: 9.A
Number: 6
Suppose is a real vector space and . Prove that is invertible if and only if is invertible.

Page: 300
Section: 9.A
Number: 7
Suppose is a real vector space and . Prove that is nilpotent if and only if is nilpotent.

Page: 300
Section: 9.A
Number: 8
Suppose and are eigenvalues of . Prove that has no nonreal eigenvalues.

Page: 300
Section: 9.A
Number: 9
Prove there does not exist an operator such that is nilpotent.

Page: 300
Section: 9.A
Number: 10
Give an example of an operator such that is nilpotent.

Page: 300
Section: 9.A
Number: 286
CHAPTER Operators on Real Vector Spaces

Page: 300
Section: 9.A
Number: 11
Suppose is a real vector space and . Suppose there exist such that . Prove that has an eigenvalue if and only if .

Page: 300
Section: 9.A
Number: 12
Suppose is a real vector space and . Suppose there exist such that and is nilpotent. Prove that has no eigenvalues.

Page: 300
Section: 9.A
Number: 13
Suppose is a real vector space, , and are such that . Prove that null has even dimension for every positive integer .

Page: 300
Section: 9.A
Number: 14
Suppose is a real vector space with . Suppose is such that is nilpotent. Prove that .

Page: 300
Section: 9.A
Number: 15
Suppose is a real vector space and has no eigenvalues. Prove that every subspace of invariant under has even dimension.

Page: 300
Section: 9.A
Number: 16
Suppose is a real vector space. Prove that there exists such that if and only if has even dimension.

Page: 300
Section: 9.A
Number: 17
Suppose is a real vector space and satisfies . Define complex scalar multiplication on as follows: if , then (a) Show that the complex scalar multiplication on defined above and the addition on makes into a complex vector space. (b) Show that the dimension of as a complex vector space is half the dimension of as a real vector space.

Page: 300
Section: 9.A
Number: 18
Suppose is a real vector space and . Prove that the following are equivalent: (a) All the eigenvalues of are real. (b) There exists a basis of with respect to which has an uppertriangular matrix. © There exists a basis of consisting of generalized eigenvectors of .

Page: 300
Section: 9.A
Number: 19
Suppose is a real vector space with and is such that null . Prove that has at most two distinct eigenvalues and that has no nonreal eigenvalues.

Page: 309
Section: 9.B
Number: 1
Suppose is an isometry. Prove that there exists a nonzero vector such that .

Page: 309
Section: 9.B
Number: 2
Prove that every isometry on an odd-dimensional real inner product space has or as an eigenvalue.

Page: 309
Section: 9.B
Number: 3
Suppose is a real inner product space. Show that for defines a complex inner product on .

Page: 309
Section: 9.B
Number: 4
Suppose is a real inner product space and is self-adjoint. Show that is a self-adjoint operator on the inner product space defined by the previous exercise.

Page: 309
Section: 9.B
Number: 5
Use the previous exercise to give a proof of the Real Spectral Theorem (7.29) via complexification and the Complex Spectral Theorem (7.24).

Page: 309
Section: 9.B
Number: 6
Give an example of an operator on an inner product space such that has an invariant subspace whose orthogonal complement is not invariant under . [The exercise above shows that can fail without the hypothesis that is normal.] Suppose and has a block diagonal matrix with respect to some basis of . For , let be the operator on whose matrix with respect to the same basis is a block diagonal matrix with blocks the same size as in the matrix above, with in the block, and with all the other blocks on the diagonal equal to identity matrices (of the appropriate size). Prove that .

Page: 309
Section: 9.B
Number: 8
Suppose is the differentiation operator on the vector space in Exercise in Section 7.A. Find an orthonormal basis of such that the matrix of the normal operator has the form promised by .

Page: 319
Section: 10.A
Number: 1
Suppose and is a basis of . Prove that the matrix is invertible if and only if is invertible.

Page: 319
Section: 10.A
Number: 2
Suppose and are square matrices of the same size and . Prove that .

Page: 319
Section: 10.A
Number: 3
Suppose has the same matrix with respect to every basis of . Prove that is a scalar multiple of the identity operator.

Page: 319
Section: 10.A
Number: 4
Suppose and are bases of . Let be the operator such that for . Prove that

Page: 319
Section: 10.A
Number: 5
Suppose is a square matrix with complex entries. Prove that there exists an invertible square matrix with complex entries such that is an upper-triangular matrix.

Page: 319
Section: 10.A
Number: 6
Give an example of a real vector space and such that .

Page: 319
Section: 10.A
Number: 7
Suppose is a real vector space, , and has a basis consisting of eigenvectors of . Prove that . SECTION 10.A Trace

Page: 319
Section: 10.A
Number: 8
Suppose is an inner product space and . Define by . Find a formula for trace .

Page: 319
Section: 10.A
Number: 9
Suppose satisfies . Prove that

Page: 319
Section: 10.A
Number: 10
Suppose is an inner product space and . Prove that

Page: 319
Section: 10.A
Number: 11
Suppose is an inner product space. Suppose is a positive operator and trace . Prove that .

Page: 319
Section: 10.A
Number: 12
Suppose is an inner product space and are orthogonal projections. Prove that .

Page: 319
Section: 10.A
Number: 13
Suppose is the operator whose matrix is Someone tells you (accurately) that and are eigenvalues of . Without using a computer or writing anything down, find the third eigenvalue of .

Page: 319
Section: 10.A
Number: 14
Suppose and . Prove that trace .

Page: 319
Section: 10.A
Number: 15
Suppose . Prove that .

Page: 319
Section: 10.A
Number: 16
Prove or give a counterexample: if , then trace .

Page: 319
Section: 10.A
Number: 17
Suppose is such that for all . Prove that .

Page: 319
Section: 10.A
Number: 18
Suppose is an inner product space with orthonormal basis and . Prove that Conclude that the right side of the equation above is independent of which orthonormal basis is chosen for .

Page: 319
Section: 10.A
Number: 306
CHAPTER Trace and Determinant

Page: 319
Section: 10.A
Number: 19
Suppose is an inner product space. Prove that defines an inner product on .

Page: 319
Section: 10.A
Number: 20
Suppose is a complex inner product space and . Let be the eigenvalues of , repeated according to multiplicity. Suppose is the matrix of with respect to some orthonormal basis of . Prove that

Page: 319
Section: 10.A
Number: 21
Suppose is an inner product space. Suppose and for every . Prove that is normal. [The exercise above fails on infinite-dimensional inner product spaces, leading to what are called hyponormal operators, which have a welldeveloped theory.]

Page: 345
Section: 10.B
Number: 1
Suppose is a real vector space. Suppose has no eigenvalues. Prove that .

Page: 345
Section: 10.B
Number: 2
Suppose is a real vector space with even dimension and . Suppose det . Prove that has at least two distinct eigenvalues.

Page: 345
Section: 10.B
Number: 3
Suppose and . Let denote the eigenvalues of (or of if is a real vector space), repeated according to multiplicity. (a) Find a formula for the coefficient of in the characteristic polynomial of in terms of . (b) Find a formula for the coefficient of in the characteristic polynomial of in terms of . SECTION 10.B Determinant

Page: 345
Section: 10.B
Number: 4
Suppose and . Prove that .

Page: 345
Section: 10.B
Number: 5
Prove or give a counterexample: if , then .

Page: 345
Section: 10.B
Number: 6
Suppose is a block upper-triangular matrix where each along the diagonal is a square matrix. Prove that

Page: 345
Section: 10.B
Number: 7
Suppose is an -by- matrix with real entries. Let denote the operator on whose matrix equals , and let denote the operator on whose matrix equals . Prove that trace and .

Page: 345
Section: 10.B
Number: 8
Suppose is an inner product space and . Prove that Use this to prove that , giving a different proof than was given in .

Page: 345
Section: 10.B
Number: 9
Suppose is an open subset of and is a function from to . Suppose and is differentiable at . Prove that the operator satisfying the equation in is unique. [This exercise shows that the notation is justified.]

Page: 345
Section: 10.B
Number: 10
Suppose and . Prove that is differentiable at and .

Page: 345
Section: 10.B
Number: 11
Find a suitable hypothesis on and then prove .

Page: 345
Section: 10.B
Number: 12
Let be positive numbers. Find the volume of the ellipsoid by finding a set whose volume you know and an operator such that equals the ellipsoid above.